Modeling Continuous and Discrete Biomarker Outcomes

For continuous variables, we assume that the observed response, w (the y-axis variable figure 1a), is normally distributed with a mean of $h(x,\mathit{c})$ and a s.d. of $\psi (x,\mathit{\kappa })$ , $$w\hspace{0.17em}\sim \hspace{0.17em}\mathcal{N}(h,\psi ).$$ For the parametrization of the mean, we use a scaled, offset power law, $h(x,\mathit{c})=c_2{x}^{c_1}+c_3.$ For the parametrization of the s.d., we consider either constant noise, $\psi (x,\mathit{\kappa })={\kappa }_1$ (homoscedastic) or linear positive noise, $\psi (x,\mathit{\kappa })={\kappa }_1[1+{\kappa }_{2}x]$ (heteroscedastic). To ensure that the standard deviation is always positive, we require all parameters to be positive; since the intercept for the heteroscedastic model must be positive, we refer to this noise model as ‘linear positive intercept.’ Figure 1a shows the FDL value (w) as a function of age (x). The red dots are pairs of values for individuals of known age. The solid line is the function $h(x,\mathit{c})$ that resulted from a maximum likelihood, univariate fit to the known age data for the heteroscedastic model. The shaded region shows the noise level as a function of age ( $h\pm \psi$ ).
Figure 1.

(a) FDL versus known age with a heteroscedastic maximum-likelihood fit. Red dots are observations, the black line is the mean response, and the blue shaded region marks the noise bounds. (b) Maxillary first molar (max_M1) developmental score versus known age. (c) Probability of observing the dental developmental stage of 7 (v = 7) as a function of age for max_M1. The grey band that extends from the middle to bottom plot marks the range of ages for which v = 7 is observed in the data. The black curve is the predicted probability the model preferred by cross-validation (power law for the mean and heteroscedastic noise). The red dots are the observed proportions in the underlying data, which are calculated by binning observations by known age value and calculating the proportion of observations in each bin for which v = 7.